Relations Definition and 579 Threads

Hi guys I am having trouble with this question (i have attached). Any help with it would be very much appreciated. Many thanks in advance Pete

Hi :) I have my Discrete maths final in 2 days, and I was doing some practice questions and came across 2 parts that completely baffled me - I moved onto my course a bit late so I missed chunks from classes. please please please, can you explain them to me? I've put the questions in...

My problem is as follows: Prove that the set of Godel numbers of provable formulas (P_f) is recursive iff the set of Godel numbers of the provable sentences (P_s) is recursive. Now I'm given as a hint to use the fact that the relation: "E_x is a formula and E_y is its closure" is recursive...

Homework Statement I came across a slightly unusual question today. It started out fine, just asking me to derive a maxwell relation but then asked under what conditions is this relationship valid. Homework Equations The Attempt at a Solution In deriving the relation I need to assume U...

Homework Statement Check that the spin matrices (Equations 4.145 and 4.147) obey the fundamental commutation relations for angular momentum, Equation 4.134.Homework Equations Eq. 4.147a --> S_{x} = \frac{\hbar}{2}\begin{pmatrix}0 & 1 \\ 1 & 0 \end{pmatrix} Eq. 4.147b --> S_{y} =...

It's not often I'm shocked in a positive way, but could this be true? I forum where science "discussions" are at least consistantly of higher caliber then "evolution sukz because bible says so, lol"? I'm sorry if my shock confuses and annoys others on this board but imagine traveling through...

Given tanA=y/x.....(1) Can anyone tell me how you get the following relations: =>sinA=ay/sqrt(x^2+y^2).....(2) =>cosA=ax/sqrt(x^2+y^2)....(3) where a=(+/-)1 I know tanA=sinA/cosA and sin^2(A)+cos^2(A)=1...and I can see by substituting (2) and (3) into (1) it works, but I really...

Homework Statement Homework Equations Given is c_p = \sum_{n_i} (-1)^{N_p}|...,n_p=0,...><...,n_p=1,...| c_p^{\dagger} = \sum_{n_i} (-1)^{N_p}|...,n_p=1,...><...,n_p=0,...| N_p = \sum_{i=1}^{p-1}n_i Verify \{c_p,c_q\} = \{c_p^{\dagger},c_q\} = \{c_p^{\dagger},c_q^{\dagger}\} = 0...

In reviewing the derivation of the quantization of angular momentum-like operators from their commutation relations, I noticed that there is nothing a priori from which you can deduce the degeneracy of the eigenstates. While this is not a problem for angular momentum, in which other constraints...

Are there any sets A for which (\mathcal{P}(A), \subseteq) is totally ordered? Prove your answer. To be courteous, I will include the definitions for partial ordering and total ordering. A relation is a partial order if the relation is reflexive, antisymmetric, and transitive. (in this...

i need to find the commutation relation for: [x_i , p_i ^n p_j^m p_k^l] I could apply a test function g(x,y,z) to this and get: =x_i p_i ^n p_j^m p_k^l g - p_i ^n p_j^m p_k^l x_i g but from here I'm not sure where to go. any help would be appreciated.

Alright, this is probably a really redundant question but for some reason it is giving me trouble. Let's say you are given the entropy of a black hole as: S=\frac{8\pi^2GM^2k}{hc} (thanks Stephen Hawking) And you have the relation between temperature and entropy/energy \frac{1}{T}=...

Hello all, I've been thinking about the connection between commutativity of operators and uncertainty. I've convinced myself that to have simultaneous eigenstates is a necessary and sufficient condition for two observeables to be measured simultaneously and accurately. It's also clear...

I have a problem with deriving another result. Sorry I am new to this field. Please see the attached PDF - everything is there.

I am confused about completeness relations. I thought a completeness relation was something like: I = \sum_{i = 1}^n |i><i| = \sum_{i=1}^n P_i [ where P_i is the projection operator onto i. Now I saw this called a completeness relation as well: \delta(x - x') = \sum_{n=0}^\infty...

I know acoustic and optical phonons can interact with one another. Also, longitudinal and transverse phonons can interact with one another. I am wondering can a longitudinal phonon in one plane act with a transverse phonon from another plane to create a third phonon? Or, do these...

Homework Statement Explain why each of the following relations is not a functions for all reals. a. f\left( x \right) = \frac{1}{{x - 5}} b. f\left( x \right) = \frac{{1 + 2x}}{{1 + 5x}} c. f\left( x \right) = \sqrt {x + 2} how would i do this? and why is it so? many thanks...

I am trying to understand 3D Si dispersion relations and reciprocal lattice vectors. My confusion is that when I look at dispersion relations the wave vector typically is normalized from 0 to 1 by a/2pi. I thought the edge of the first BZ was pi/a. Is this correct or is it 2pi/a for a diamond...

Homework Statement A is some set. R is a relation (set of ordered pairs), and is transitive on A. S = {(x,y) | (x,y) is element of R, (y,x) is not element of R} Show that S is transitive and trichotomic on A. Homework Equations Transitivity: With xRy and yRz ==> xRz The...

Homework Statement Prove that if R is a symmetric relation on A, and Dom(R) = A, then R = the identity relation. 2. The attempt at a solution My problem is... I don't believe the claim. At all. If A = {1, 2, 3} and R = {(1, 2), (2, 1), (3, 1), (1, 3)}, that satisfies the antecedent, and...

Homework Statement Here's my problem - Give the order of linear homogeneous recurrence relations with constant coefficients for: An = 2na(n-1) The Attempt at a Solution I have no idea on how to start this problem - Any help would be greatly appreciated.

given that [x,p]=ih, show that if x=x, p has the representation p=-iħ∂/∂x+f(x) where f(x) is an arbitrary function of x

Homework Statement On the set of Natural Numbers from 1 to 10000 are given the following identity relations. R1 ; n R1 m where m and n have the same remainder by division by 24, that is mod n 24 == mod m 24. R2 ; n R2 m where n and m have in decimal notation the same number of 2s R3; n...

Homework Statement The canonical commutation relations for a particl moving in 3D are [\hat{x},\hat{p_{x}}]= i\hbar [\hat{y},\hat{p_{y}}]= i\hbar [\hat{z},\hat{p_{z}}]= i\hbar and all other commutators involving x, px, y ,py, z , pz (they should all have a hat on eahc of them signifying...

Hello everyone. This problem has a few parts, and I'm on the last part and I'm having troubles and im' guessing my way is not the correct method. But here is the question. Let A be a set with 8 elements. a. how many binary relations are there on A? answer: A binary relation is any...

Homework Statement I need to express V in terms of Z and A. i know V= (3/5)Y + (2/5)Z given y=2A-3X X=(1/6)Z + (5/6)Y Homework Equations The Attempt at a Solution ok. my first attempt was to isolate y in the second equation, and let the two equations equal...

The cauchy Riemann relations can be written: \frac{\partial f}{\partial \bar{z}}=0 Is there an 'easy to see reason' why a function should not depend on the independent variable [itex]\bar{z}[/tex] to be differentiable?

I know that the functions e^{2 \pi inx} for n \in \mathbb{Z} are a base in the space of functions whith period 1. How do I derive the orthogonality relations for these functions?

I was wondering, what is the difference between an operator and a relation? For example, instead of saying 2+3 I can say Add(2,3). Or the \frac{df(x)}{dx} operator can be written as D(f(x)). I fail to see any difference between an operator and a relation. What do you guys think?

Does the usual commutation relations, e.g. between position and momentum, remains valid in relativistic quantum mechanics?

The Kramers-Kronig relations allows one to calculate the real part of the permitivity knowing the imaginary part or vice-versa: http://en.wikipedia.org/wiki/Kramers-Kronig_relations But in what situation will one know either the imginary part but not the real part or the real part but not the...

Is there a commonly-used name for a complete preorder (a transitive and total relation, Sloan's A000670 and A011782 for labeled and unlabeled, respectively) within set theory? (Not a total order, mind you -- it need not be antisymmetric.) I've heard the term "weak order", but that's from the...

-Let's suppose we have 2 gases ..one is a "Fermi" gas under an Harmonic potential and the other is a "Bose" gas under another Harmonic potential... in both cases (as an approximation) the particles (bosons and electrons are Non-interacting) then we could write the partition functions. \prod...

I am reading the first chapter of Sakurai's Modern QM and from pages 30 and 32 respectively, I understand that (i) If [A,B]=0, then they share the same set of eigenstates. (ii) Conversely, if two operators have the same eigenstates, then they commute. But we know that [L^2,L_z]=0...

Question: "Find a recurrence relation and initial conditions for the sequence {a sub n} if a sub n is the number of bit strings of length n that contain three consecutive 0's." So here's what I have so far... n > 3 n = 4, 1000, 0001 n = 5, 10000, 00001, 00010, 01000, 10001 n = 6...

Since we have anticommuting relations for the quantum Dirac fields, this will bring us to the similar classical correspondance but result in Grassmann spinor field function instead. (such as path integral) So when we consider an arbitrary interaction term that like (\bar{\psi} \psi)^n, if...

Okay, so I have a homework problem I'm a little confused about, The textbook is pretty useless and we didn't go into types of orders very much in class. So, am I to show that the dictionary order is reflexive, antisymmetric, and transitive on XxX, since XxX is already linearly ordered? I...

Ok; this is another thread that covers two questions. I didn't want to mix them with my previous post; it's from the same 'section' but the questions are different. If any mods have issues with this, please say so. 1) If R \cup S is reflexive, then either R is reflexive or S is reflexive...

Ok so here's one of the questions we've been assigned... So I can graphically see what this relation looks like, and from that I've shown it's reflexive. Now I'm working on proving it as being symmetric, but I can't put it into words. b) ~ is symmetric. Well we want to show that aRb ->...

Relations Again :( K, so I am studying for the upcoming midterm... and their is this question in the book... Let A = {a} and B = {1,2,3}. List all the possible relations between A and B. So Ordered pairs, Cartesian Products and Relations are all together in the chapter, and I am really...

ok i don't know why i can't grasp this and i feel so stupid... here's an example in the book which i do get... Let S denote the set of all nonempty subsets of {1, 2, 3, 4, 5}, and define a R b to mean that a \cap b not equal to \emptyset. The R is clearly reflexive and symmetric...

R = the real numbers A = R x R; (x,y) \equiv (x_1,y_1) means that x^2 + y^2 = x_1^2 + y_1^2; B= {x is in R | x>= 0 } Find a well defined bijection sigma : A_\equiv -> B like the last problem, I just can't seem to find the right way to solve this??

Z = all integers A = Z; m is related to n, means that m^2 - n^2 is even; B = {0,1} I already proved that this is a equivalence relation, but i just don't know how to; I need to find a well defined bejection sigma : A_= -> B I hope this makes sense.. i wrote it up as well as I...

I'm doing this problem in the book - their are 2 of this kind and they have no answers in the back.. so i thought ill post one. Let S be the Cartesian coodinate plane R x R and define a relation R on S by (a,b)R(c,d) iff a+d=b+c. Verify that R is an equivalence relation and describe the...

The antisymmetric relations on a set {a,b} are those, which do not contain both of the pairs (a,b) and (b,a) because that would imply a = b, however a can't equal b since they are elements of a set? PS: In our course we allow only one copy of an element in a set, so {a,b} is a set only if a...

I answered this wrong on a test, but now I've come up with a different solution. Problem: Prove that a relation xRy\Leftrightarrow x-y\in\mathbb{Z} defined on \mathbb{R} is an equivalence relation. Solution: 1.) Reflexivity: xRx,\forall x\in\mathbb{R} For every x we have x-x=0 which is an...

Ok I'm giving these another go. I found the following DE from a reduction of order problem and figured that it would be an alright question if I turned it into one requiring a series solution. However I'm stuck. I think it's just a matter of index shifts to get an appropriate recurrence relation...

Can anyone just check if I got it right please? And if so could you just explain the theorems that come with each line? Many many thanks in advance :smile: (A-B) n (B-A) = (AuB’) n (BuA’) = (Au(BuA’)) u(B’n (BuA’)) = ((AnB) u (AnA’)) u ((B’nB) u (B’nA’)) = (AnB) u Ø u Ø u (B’nA’) =...

Let AxB be the set of ordered pairs (a,b) where a and b belong to the set of natural numbers N. A relation p: AxB -> AxB is defined by: (a,b)p(c,d) <-> a+d = b+c (i) Is (2,6) related to (4,8)? Give three ordered pairs which are related to (2,6) ANSWER: Yes (2,6)p(4,8) as 2+8 = 6+4 =...

What are the conditons on A, B and C for (AuB)nC = Au(BnC) ? Is it that AnBnC ? Can someone explain if they are different and why? :confused: Now If A = {irrationals}, B= {integers} and C={reals} does the equality from above hold in this case? I answered yes the equality holds as...